Chasing Starlight 1: Sketching with Feynman

by Shane L. Larson

Some science concepts are so fundamental that they’ve become summarized by pithy soundbites that have invaded mainstream culture. For instance, have you ever wondered where the phrase “opposites attract” comes from? It’s a substantive scientific concept that has become so ingrained in our culture as to become a meme: from Lady and the Tramp, to Claire and John in The Breakfast Club, to dogs and cats being friends.

"Opposites attract" is such a common phrase in our culture that it is meme unto itself in the movies. (L) Disney's 1955 classic, "Lady & the Tramp". (R) Claire and John in the 1985 classic, "The Breakfast Club."

“Opposites attract” is such a common phrase in our culture that it is meme unto itself in the movies. (L) Disney’s 1955 classic, “Lady & the Tramp“. (R) Claire and John in the 1985 classic, “The Breakfast Club.”

It comes from physics, from an area of study called “electromagnetism.” Our understanding of how electromagnetism works is built around the study of fundamental particles called “charges.” Charges come in two flavors that we call positive and negative. The most common charge you and I encounter every day is called the electron, which is negative in flavor.

The phrase “opposites attract” is about charges — opposite charges are attracted to one another (and like charges are repulsed by other like charges). So who is the electron attracted to? In principle it is attracted to any positive charge it encounters. The most common positive charge that we all encounter in our everyday lives is called the proton. From our perspective both the electron and the proton are tiny particles, but the proton is about 1800 times more massive than the electron; that’s about the same as you compared to a space shuttle. So what happens when they see each other? They experience an attractive force, because they are charged.  The proton doesn’t move much, because it is so massive, so the electron falls toward the proton. Pulled inexorably together, they bind together and form a hydrogen atom — a proton, with an attendant electron captured in the cloud of the proton’s influence.

Expected forces between electrical charges. (Top) Two positive charges repel each other. (Mid) Two negative charges repel each other. (Bottom) Opposite charges attract.

Expected forces between electrical charges. (Top) Two positive charges repel each other. (Mid) Two negative charges repel each other. (Bottom) Opposite charges attract.

There are other positive particles in Nature, not as plentiful as the proton. One of those is called the positron. The positron has excactly the same properties as the electron, except for the flavor of its charge: the positron is positive. Otherwise it has the same mass, the same size, the same behaviour of the electron. The positron is the anti-matter twin of the electron. What happens when a positron and an electron approach each other? Just as with the proton, there is an attractive force, but since they are the same mass, both the electron and positron fall toward one another. They are so small they very likely don’t hit one another, but their close encounter deflects them onto new paths; physicists call this “scattering.”

If they do hit each other, they completely destroy each other, converting from particles into light. This process, where matter-antimatter particle pairs destroy one another is called “annihilation.”  If two electrons approach each other, they feel a repulsive force which deflects this onto new paths; this is also scattering.

If identical matter and anti-matter particles collide (like an electron and a positron) they annihilate, creating light with an energy equal to the mass energy of the two particles.

If identical matter and anti-matter particles collide (like an electron and a positron) they annihilate, creating light with an energy equal to the mass energy of the two particles.

So how do physicists describe what is going on between charges? As with all things in science, there are many ways of describing and talking about any physical phenomena. Just using words, telling a story that says what happens, maybe with a few pictures is a time honored way to introduce complex ideas to anyone. This is basically what we are doing here in this blog.

But eventually, people begin to start thinking about what all these awesome facts of Nature might be useful for. Charges repel and attract — can I do anything with that? Can I harness this physical effect and either make something that is entertaining, might make me money, or better yet, can be used to improve our lives? The answer is yes to all of these questions! Of course you can. But in order to harness Nature, in order to make something you need to have predictive power. What does that mean? It means if you know everything about some physical system — a pool of water, a pile of sludge, a rock on a table, or a cloud of electroms — then you can predict what will happen to that system in the future as it interacts with the world around it.

You have learned the predictive power of science at a very young age, and use it every day. What happens if you set your hot cup of coffee on the counter and leave it there for 3 hours? It cools off — you know that. What happens if you throw a baseball as hard as you can straight up in the air? It falls back to the ground — you know that. What happens if you hit an egg with a hammer? It makes a big mess (which you should clean up before your Mom/wife finds it) — you know that.

Example scenarios where your everyday experience has taught you the Laws of Physics. You understand these situations well enough that you have predictive power.

Example scenarios where your everyday experience has taught you the Laws of Physics. You understand these situations well enough that you have predictive power.

But sometimes you want to know something precise.  How long will it take my coffee to be about 105 degrees F, the perfect swigging temperature? What is the maximum height my baseball will reach, and is it enough to knock my keys out of that tree (don’t ask)?  Would that egg have still gotten on the ceiling if we had vaulted ceilings?  When you need precise predictive power, scientists turn to mathematics to express the “Laws of Nature.”

The mathematical expression of the force between two charges was first written down by a French physicist known as Charles-Augustin de Coulomb; appropriately enough that law is now called the “Coulomb Law.” His law showed something that had been empirically measured, namely that the force between two charges depended on two things:

  • The sizes of the charges involved (physicists use the symbol “q” to mean charge in an equation)
  • The force is weaker with greater distance, and in fact grows weaker with the square of the distance — if you double the distance, the force is 1/4 as large; if you triple the distance, the force is 1/9 as large (physicists use the symbol “r” to mean distance in an equation)
Inverse square laws from Nature; note the similarities. (L) Universal Law of Gravitation, describing force between two masses. (R) Coulomb Law, describing force between two charges.

Inverse square laws from Nature; note the similarities. (L) Universal Law of Gravitation, describing force between two masses. (R) Coulomb Law, describing force between two charges.

One of the most remarkable realizations that one has when seeing the Coulomb law for the first time is that it has exactly the same form as another great law of physics, Newton’s Universal Law of Gravitation. This is a hint that there are deep mysteries afoot, that the awesome machinery of Nature has a secret and is giving us a casual, teasing glimpse under the hood.  Physicists call this the “inverse square law.”

These kinds of teasing glimpses make scientists dig deeper, to try and better understand how the Cosmos is put together. Much of the middle of the Twentieth Century was spent trying to understand the structure and interactions of matter on the tiniest scales. The outcome of those herculean efforts of thought and experiment was a list of the particles that make up all matter and the rules for how they interact together — it is called “The Standard Model,” and it is one of the most successful descriptions of Nature humans have ever discovered. It is a triumph of our intellect, a testament to our ingenuity and diligence.

Richard Feynman.

Richard Feynman.

As you might imagine, the mathematical structure of the Standard Model is intimidating to witness, even if you’ve been trained in the field! But luckily for us, there is a beautiful and ingenious shorthand that summarizes the mathematical rules of how particles interact with one another, called Feynman Diagrams. Developed by the indefatigable Richard Feynman, the diagrams can quickly be sketched, and completely encode the content of the mathematical equations. The beauty of the method is that when you draw the diagrams, you can imagine the particles as if they were little ping-pong balls bouncing off of one another.

The basic vertex used to build Feynman diagrams.

The basic vertex used to build Feynman diagrams.

Let’s restrict our attention to just three particles: electrons, positrons, and photons.  In a diagram, an electron is represented by an arrow that generally points toward the top of your screen, a positron is represented by an arrow that generally points toward the bottom of your screen, and a photon is represented by a squiggly line.  The fundamental building block of a Feynman diagram is called a vertex. Interactions are represented by linking vertices together. The rules for drawing and interpreting basic Feynman Diagrams are as follows:

  • An interaction between two particles must have an even number of vertices
  • You can rotate and move the lines going into a vertex any way you like, so long as there is one arrow pointing into the vertex and one pointing out of the vertex.
  • When linking two vertices together, photons connect to photons, and arrows (electrons and positrons) cannot point against each other.

How do you draw and interpret Feynman Diagrams? Let’s consider a specific example to talk about that. From the Coulomb law, we know that opposite charges repel each other. That means if two electrons fly toward one other, the force between them should grow until it pushes them apart.  This is where the physical interpretation of the Feynman Diagrams are useful — I know initially two electrons should be rushing toward each other, they will interact somehow (this will be represented by the vertices), then at the end they should be rushing apart.  If I draw a Feynman Diagram representing this, and cover up all the bits with the vertices with a sticky note, what I should see is two arrows pointing inward at the bottom of my diagram, and two arrows pointing outward at the top of the diagram.

(L) The physical motion of two electrons that scatter off of each other. They approach, but feel a force pushing them apart, so eventually they change the direction of travel. (R) A Feynman diagram for this process looks very similar. Under the sticky note is "in the fog" --- necessary bits that construct the proper mathematical description of the scattering, but is not observable in the same sense as the particles not covered by the sticky note.

(L) The physical motion of two electrons that scatter off of each other. They approach, but feel a force pushing them apart, so eventually they change the direction of travel. (R) A Feynman diagram for this process looks very similar. Under the sticky note is “in the fog” — necessary bits that construct the proper mathematical description of the scattering, but is not observable in the same sense as the particles not covered by the sticky note.

So what’s under the sticky note? Under the sticky note is any possible way you can imagine to link the lines together with vertices.  The most fundamental interaction is two vertices, with a photon linking them.  When we see this diagram, we “read it” and say “the two electrons interacted by exchanging a virtual photon.”  In the language of the Coulomb law, we said there was an electromagnetic force between the electrons. In the language of the Standard Model and the Feynman Diagrams we say that the “photon is the electromagnetic force carrier.”

The most basic Feynman diagram showing electron scattering, with two vertices.

The most basic Feynman diagram showing electron scattering, with two vertices.

So does that mean there are photons flying all over the place each time electric charges repulse or attract each other?  Not exactly. We say that the vertices are “in the quantum fog” — the exchange between vertices is a completely unobservable aspect of the interaction. That may seem a bit disconcerting, but I usually take comfort in the fact that the diagrams are not physical representations of the interaction, but are rather a clever code for some very involved mathematics.

An example alternative Feynman diagram for electron scattering, with 4 vertices.

An example alternative Feynman diagram for electron scattering, with 4 vertices.

If you spend some time tinkering with the diagrams, you may notice that there is more than one way of representing the interaction — there is more than one way to draw the Feynman diagram! That is part of the mathematics; each diagram has a slightly different mathematical interpretation, and the full calculation of the outcome of an interaction depends on knowing all the possible different ways of drawing the diagrams! The great utility of the Feynman Diagram method is your job as a physicist is reduced to drawing diagrams. Instead of writing all the algebra out longhand, you just have to be creative and imaginative enough to figure out what every single diagram is.

So why have I told you all of this? Because one of the great truths of science is that when you write down the Laws of Nature, fiddling with those laws will often lead to the discovery of something previously unknown.  Consider the following diagram.

The simplest Feynman diagram representing "Delbruck scattering" --- the scattering of a photon off another photon.

The simplest Feynman diagram representing “Delbruck scattering” — the scattering of a photon off another photon.

If I put a sticky note over the vertices, what do I see? This is two photons approaching one another and scattering off of one another! This is called “Delbruck scattering.”

Here’s another interesting diagram where two photons come together and utterly vanish.

(L) Pair production --- two photons collide and vanish, and in their place leave behind a positron and an electron. (R) Two Feynman diagrams showing this process.

(L) Pair production — two photons collide and vanish, and in their place leave behind a positron and an electron. (R) Two Feynman diagrams showing this process.

If I put a sticky note over the vertices, what do I see? This is two photons approaching one another and vanishing! In their stead, two particles — one electron and one positron — spontaneously appear! This is called “pair creation” and is important in many different instances in physics and astronomy.

Generically, both of these diagrams are light scattering off of light.  It seems to be far outside of our normal everyday experience. If you and I shine flashlights at each other, the beams fly right through each other. But the laws of physics allow the diagrams I’ve drawn above. Amazingly, if we go into our labs and particle accelerators and look for these effects, we see them! This kind of validation gives us confidence that our understanding of how the world works is on the right track. It’s also exhilarating — it makes me a little giddy some days when it strikes me that we were able to figure this out. Such is the awesome power afforded to us by brains.

So light scatters off of light.  Of what possible use could that be? Well, as it turns out, it can be used to discover ancient starlight. This will be the subject of our next chat.

————————————

This post is the first part in a sequence talking about clever ways to use starlight to understand the Cosmos. The second part is here.

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